effect of magnetic field on steady boundary layer slip flow along with heat … · 2017. 3. 9. ·...

Global Journal of Pure and Applied Mathematics.

ISSN 0973-1768 Volume 13, Number 2 (2017), pp. 647-661

© Research India Publications

http://www.ripublication.com

Effect of Magnetic Field on Steady Boundary Layer

Slip Flow Along With Heat and Mass Transfer over a

Flat Porous Plate Embedded in a Porous Medium

K. Sumathi1, T. Arunachalam2 and R. Kavitha3

1Department of Mathematics, P.S.G.R Krishnammal College for Women, Coimbatore - 641 004, India.

2Department of Mathematics, Kumaraguru College of Technology, Coimbatore - 641049, India.

3Department of Mathematics, P.S.G.R. Krishnammal College for Women, Coimbatore - 641004, India.

Abstract

In this study, we investigate the effect of magnetic field on steady boundary

layer flow along with heat and mass transfer over a flat porous plate with slip

boundary condition at the boundary in a porous medium using shooting

method with fourth order Runge-Kutta Method. The velocity, temperature and

concentration distribution are calculated for different governing parameters.

The effect of various non-dimensional parameters like magnetic parameter

(M), Schmidt number (𝑆𝑐), Soret number (𝑆𝑟), Suction/blowing parameter (𝑆) are investigated with the aid of graphs. The effect of these dimensionless

parameters skin-friction, rate of heat and mass transfer are discussed in detail

to understand the physical aspects of the solutions.

Keywords: MHD, Heat and Mass transfer, Porous medium, Steady flow,

Nusselt number, Sherwood number, Soret number

mailto:[email protected]:[email protected]:[email protected]

648 K. Sumathi T. Arunachalam and R. Kavitha

INTRODUCTION

The heat and mass transfer of viscous incompressible fluid flow over a porous plate in

the presence of magnetic field has tremendous applications in many engineering fields

such as heat exchanger devices, petroleum reservoirs, chemical catalytic reactors and

processes, geothermal and geophysical engineering, aerodynamic engineering and

others.

The applications of convective heat transfer and fluid flow through porous medium

are found in many engineering and geophysical fields such as geothermal and

petroleum resources, solid matrix heat exchangers, filtration and purification process,

thermal insulation drying of porous solids, enhanced oil recovery, cooling of nuclear

reactors and underground water resources in agriculture engineering. In view of these

applications, many researchers have studied MHD free convective heat and mass

transfer flow through porous medium.

The effect of magnetic field on heat and mass transfer under various physical

conditions was investigated by several authors. Hydromagnetic mixed convective heat

and mass transfer over a vertical plate with a convective surface boundary condition

was studied by Aziz [4]. Freidoonimehr, Rashidi and Jailpour [5] studied the heat and

mass transfer in an incompressible steady laminar MHD flow using the combination

of differential transform method and Padé approximant. The effect of magnetic field

on free convective heat transfer was studied by Sparrow and Cess [12]. It was

observed that the free convective heat transfer to liquid metals are affected

significantly by magnetic field, but other fluids experienced negligible effects.

Andersson [1] studied the slip effects on boundary layer stagnation-point flow and

heat transfer towards a shrinking sheet and later it was extended by Santosh

Chaudhary and Pradeep Kumar [11] for viscous, incompressible, electrically

conducting fluid near a stagnation point past a shrinking sheet with slip in the

presence of a magnetic field. Anjalidevi and Kandaswamy [2] explored the effects of

magnetic field on heat and mass transfer flow along a semi-infinite horizontal plate.

Thermal radiation and non-uniform magnetic field was considered by Mohammad

Mehdi Rashidi et al. [9] who employed the homotopy analysis method. Pal and Talukdar [10] studied an unsteady MHD convective heat mass transfer in the presence

of thermal radiation and chemical reaction using perturbation method. Gangadhar [6]

studied the similarity solution of hydromagnetic heat and mass transfer over a vertical

plate with convective surface boundary condition and chemical reaction.

The effect of temperature-dependent viscosity on free convective flow past a vertical

porous plate was studied by Makinde [7,8] in the presence of a magnetic field without

heat source and with heat source. Asim Aziz et al. [3] studied the steady boundary layer slip flow along heat and mass transfer in porous medium.

The absence of magnetic field in Asim Aziz et al. [3] motivated us to take up the

Effect of Magnetic Field on Steady Boundary Layer Slip Flow… 649

present work wherein we study, the steady boundary layer slip flow along with heat

and mass transfer over a flat porous plate embedded in a porous medium in the

presence of magnetic field. In order to examine the accuracy of our result, the effect

of various parameter on velocity, temperature and concentration are calculated and

plotted. The results obtained are validated for vanishing magnetic field which is same

as the result obtained by Asim Aziz et al. [3].

FLOW DESCRIPTION AND GOVERNING EQUATIONS

We consider a steady two dimensional laminar flow of a viscous incompressible

electrically conducting fluid with heat and mass transfer over a porous flat plate in the

presence of transverse magnetic field embedded in a porous medium. The x-axis is

taken along the plate and the y-axis is taken normal to the plate [see figure 1]. In

order to simplify the model, all body forces are neglected and we assume that the fluid

flow is stable.

Under these assumptions, the governing equations for the continuity, momentum and

energy are taken in the following form.

𝜕𝑢

𝜕𝑥+

𝜕𝑣

𝜕𝑦= 0 (1)

𝑢𝜕𝑢

𝜕𝑥+ 𝑣

𝜕𝑢

𝜕𝑦= 𝛾

𝜕2𝑢

𝜕𝑦2−

𝛾

𝑘(𝑢 − 𝑈∞) +

𝜎𝐵02

𝜌(𝑈∞ − 𝑢) (2)

𝑢𝜕𝑇

𝜕𝑥+ 𝑣

𝜕𝑇

𝜕𝑦=

𝜅

𝜌𝐶𝑝

𝜕2𝑇

𝜕𝑦2 (3)

𝑢𝜕𝐶

𝜕𝑥+ 𝑣

𝜕𝐶

𝜕𝑦= 𝐷𝑀

𝜕2𝐶

𝜕𝑦2+ 𝐷𝑇

𝜕2𝑇

𝜕𝑦2 (4)


Figure 1. Schematic diagram of the problem

In the above system of equations, 𝑢 and 𝑣 represent velocity components in 𝑥 and 𝑦

directions respectively. 𝑇, 𝐶, 𝜇, 𝜌, 𝜎 and 𝛾 = (𝜇 𝜌) ⁄ represent respectively the

temperature, mass concentration, coefficient of viscosity, density, electrical

conductivity and kinematic viscosity of the fluid. The constant parameters in the

system are: 𝑘, 𝐶𝑝, 𝜅, 𝐷𝑀 and 𝐷𝑇 respectively the permeability of porous material,

specific heat at constant pressure, thermal conductivity of the fluid, molecular

diffusivity and thermal diffusivity. �⃗� (𝑥) is the magnetic field in the 𝑦-direction and is

given by �⃗� (𝑥) = 𝐵0 (𝑥)1 2⁄⁄ .

The appropriate boundary conditions for velocity, temperature and mass concentration

are given by,

𝑢 = 𝐿1 (𝜕𝑢

𝜕𝑦), 𝑣 = 𝑣𝑤 at y=0; 𝑢 → 𝑈∞ as 𝑦 → ∞, (5)

𝑇 = 𝑇𝑤 + 𝐷1 (𝜕𝑇

𝜕𝑦) at 𝑦 = 0; 𝑇 → 𝑇∞ as 𝑦 → ∞ , (6)

𝐶 = 𝐶𝑤 + 𝑁1 (𝜕𝐶

𝜕𝑦) at 𝑦 = 0; 𝐶 → 𝐶∞ as 𝑦 → ∞ , (7)


In equations (5)-(7) 𝐿1 = 𝐿(𝑅𝑒𝑥)1 2⁄ is the velocity slip factor, 𝐷1 = 𝐷(𝑅𝑒𝑥)

1 2⁄ is the

thermal slip factor and 𝑁1 = 𝑁(𝑅𝑒𝑥)1 2⁄ mass slip factor. Here 𝐿, 𝐷 and 𝑁 are the

initial values of velocity, thermal and mass slip factors respectively and 𝐿, 𝐷 and 𝑁 all

three have dimensions of length. Moreover,𝑈∞, 𝑇∞ and 𝐶∞ denote free stream

velocity, free stream temperature, and free stream mass concentration respectively. In

these equations 𝑇𝑤 and 𝐶𝑤 represent the temperature and mass concentration of the

plate respectively. The velocity 𝑣𝑤 defines suction or blowing through the porous

plate and is written as 𝑣𝑤 = 𝑣0 √𝑥⁄ . In this relation 𝑣0 is constant linked with suction

if 𝑣0 < 0 and blowing when 𝑣0 > 0.

We introduce the stream function 𝜓(𝑥, 𝑦) as

𝑢 =𝜕𝜓

𝜕𝑦 , 𝑣 = −

𝜕𝜓

𝜕𝑥 (8)

SOLUTION OF THE PROBLEM

In this work, similarity technique is used to solve the system of equations (1)-(4)

along with the boundary conditions (5)-(7). The similarity transformations are,

𝜓 = √𝑈∞𝛾𝑥𝑓(𝜂), 𝜃(𝜂) =𝑇−𝑇∞𝑇𝑤−𝑇∞

, 𝜙(𝜂) =𝐶(𝑥)−𝐶∞

(𝐶𝑤−𝐶∞)𝑥, (9)

where 𝜂 is similarity variable defined as 𝜂 = √𝑅𝑒𝑥(𝑦 𝑥⁄ ).

Introducing the above transformations in equations (1)-(4), we obtain the system of

ordinary differential equations,

𝑓′′′ +1

2𝑓𝑓′′ − 𝑘∗(𝑓′ − 1) − 𝑀(𝑓′ − 1) = 0, (10)

𝜃′′ +1

2𝑃𝑟𝑓𝜃′ = 0, (11)

𝜙′′ − 𝑆𝑐𝑓′𝜙 +1

2𝑆𝑐𝑓𝜙′ + 𝑆𝑐𝑆𝑟𝜃′′ = 0 (12)

In equation (10), 𝑘∗ = 1 𝐷𝑎𝑥𝑅𝑒𝑥⁄ represents the permeability of porous medium and

𝐷𝑎𝑥 = 𝑘 𝑎2 = 𝑘0 𝑥⁄⁄ is the local Darcy number. Here 𝑘0 is the constant, 𝑀 =

𝜎𝐵02𝑥

𝜌𝑈∞ is

the magnetic parameter. In equation (11) 𝑃𝑟 = 𝜇𝐶𝑝 𝜅⁄ is the Prandtl number. Finally

in equation (12), 𝑆𝑐 =𝛾

𝐷𝑚 is the Schmidt number and

(𝑇𝑤−𝑇∞)𝐷𝑇

(𝐶𝑤−𝐶∞)𝑥𝛾 is the Soret number.

The boundary conditions (5)-(7) transform to the following form,

𝑓(𝜂) = 𝑆, 𝑓′(𝜂) = 𝛿𝑓′′(𝜂) at 𝜂 = 0; 𝑓′(𝜂) → 1 as 𝜂 → ∞ (13)

𝜃(𝜂) = 1 + 𝛽𝜃′(𝜂) at 𝜂 = 0; 𝜃(𝜂) → 0 as 𝜂 → ∞ (14)


𝜙 = 1 + 𝜍𝜙′ at 𝜂 = 0; 𝜙 → 0 as 𝜂 → ∞ (15)

where 𝑆 = (−2𝑣𝑤 𝑈∞)(𝑅𝑒𝑥)1 2⁄ = −2𝑣0 (𝑈∞𝛾)

1 2⁄⁄⁄ , 𝑆 > 0 (i.e.𝑣0 < 0) corresponds

to suction and 𝑆 < 0 (i.e.𝑣0 > 0) corresponds to blowing, 𝛿 = 𝐿𝑈∞/𝛾 is the velocity

slip parameter, 𝛽 = 𝐷𝑈∞ 𝛾⁄ is the thermal slip parameter and 𝜍 = 𝑁𝑈∞ 𝛾⁄ is a mass

slip parameter.

The nonlinear coupled ordinary differential equations (10)-(12) subject to boundary

conditions (13)-(15) are reduced to a system of first order ordinary differential

equations as follows,

𝑓′ = 𝑤, 𝑤′ = 𝑣, 𝑣′ = −0.5𝑓𝑣 + 𝑘∗(𝑤 − 1) + 𝑀(𝑤 − 1), (16)

𝜃 ′ = 𝑧, 𝑧 ′ = −0.5𝑃𝑟𝑓𝑧, (17)

𝜙′ = 𝑒, 𝑒 ′ = 𝑆𝑐𝑤𝑏 − 0.5𝑆𝑐𝑓𝑒 − 𝑆𝑐𝑆𝑟𝑧 ′ (18)

and boundary conditions become,

𝑥(0) = 𝑆, 𝑤(0) = 𝛿𝑣(0), 𝑦(0) = 1 + 𝛽𝑧(0), 𝑏(0) = 1 + 𝜍𝑒(0) at 𝜂 = 0 (19)

𝑤(0) → 1, 𝑦(0) → 0, 𝑏(0) → 0 as 𝜂 → ∞ (20)

We have solved the equations (16)-(18) with boundary conditions (19) and (20) using

shooting method.

The major physical quantities - the skin-friction coefficient 𝐶𝑓 , the local Nusselt

number 𝑁𝑢𝑥, and the local Sherwood number 𝑆ℎ𝑥 are defined respectively as follows,

Skin-friction Co-efficient: 𝐶𝑓 = 𝜏𝑤

𝜌𝑈𝑤2 𝑥

Local Nusselt Number: 𝑁𝑢𝑥 =𝑥𝑞𝑤

𝑘(𝑇𝑤−𝑇∞)

Local Sherwood Number: 𝑆ℎ𝑥 = 𝑥𝑞𝑚

𝐷𝐴(𝐶𝑤−𝐶∞)

where 𝜏𝑤is the skin-friction, 𝑞𝑤 is the heat flux, 𝑞𝑚 is the mass flux at the surface

respectively.

𝜏𝑤 = 𝜇 (𝜕𝑢

𝜕𝑦)𝑦=0

, 𝑞𝑤 = −𝑘 (𝜕𝑇

𝜕𝑦)𝑦=0

, 𝑞𝑚 = −𝐷𝐵 (𝜕𝐶

𝜕𝑦)𝑦=0

Applying the non-dimensionless transformations (9) and we obtain,

𝑓′′(0) = 𝐶𝑓(𝑅𝑒𝑥)1/2,

−𝜃′(0) = 𝑁𝑢𝑥(𝑅𝑒𝑥)−1/2 ,

−𝜙′(0) = 𝑥𝑆ℎ𝑥(𝑅𝑒𝑥)−1/2

where 𝑅𝑒𝑥 = 𝑥𝑈∞

𝛾 is a local Reynolds number.


RESULTS AND DISCUSSION

In order to get physical insight of the problem we have studied the velocity,

temperature and concentration against various parameters such as magnetic parameter

M, the velocity slip parameter 𝛿, thermal slip parameter 𝛽, mass slip parameter 𝜍 ,

Schmidt number 𝑆𝑐 and Soret number 𝑆𝑟. The effect of flow parameters on velocity

field, skin-friction, Nusselt number and Shearwood number are calculated

numerically and discussed with the help of graphs.

We have shown the velocity, temperature and concentration of various values of

permeability parameter k* through figures (2)-(4) for vanishing magnetic field. It can

be seen from these figures increase in permeability causes an increase in velocity,

decrease in temperature and concentration profiles for both slip and no-slip

conditions. These results are in good agreement with Asim Aziz et al. [3].

Figures (5)-(7) depict the influence of magnetic field M on the velocity, temperature

and concentration profiles for both slip and no-slip conditions. We observe that the

velocity 𝑓 ′(𝜂) along the plate increases when magnetic field increases. Consequently

temperature 𝜃(𝜂) and concentration 𝜙(𝜂) decrease when magnetic field increases for

both slip and no-slip conditions.

Figures (8)-(10) show the variation in shear stress𝑓′′(𝜂) , the rate of heat and mass transfer for several values of magnetic parameter M. We observe that the increase in

magnetic parameter M causes increase in shear stress 𝑓′′(𝜂) and decrease in the rate of heat and mass transfer.

Figures (11)-(16) explain about the effect of slip parameter on the velocity,

temperature and concentration profiles, shear stress, and the rate of heat and mass

transfer. It can be inferred from these graphs that increase in velocity slip parameter

results in the enhancement of velocity and the rate of mass transfer while, the shear

stress, temperature, concentration and the rate of heat transfer decrease due to the

increase of slip parameter .

Figures (17) and (18) represent the effect of slip parameter on temperature and the

rate of heat transfer. We can observe from these graphs that the increase in slip

parameter results in decrease in temperature and increase in rate of heat transfer.

Figures (19) and (20) show the effect of slip parameter 𝜍 on concentration and the rate

of mass transfer. We can observe that the increase in slip parameter 𝜍 results in

decrease in concentration and increase in rate of mass transfer.

Figures (21)-(23) depict that the variation in velocity, temperature and concentration

profiles for different values of suction/blowing parameter S. Here, S>0 shows the

suction and S0, fluid velocity

increases, which in turn decreases both the fluid boundary layer and the thickness of


momentum boundary layer. Due to suction of the fluid particles at the porous wall,

velocity profile increases. In the case of blowing S0, fluid particles come close to the porous wall, due to

this temperature profile decreases, which in turn decreases the thermal boundary

layer. When suction is increased i.e., S>0, the mass concentration decreases. An

opposite phenomena is observed for blowing.

Figures (24) and (25) explain the effect of Schmidt number (𝑆𝑐) and Soret

number(𝑆𝑟) on concentration profiles. These two figures illustrate that the increase in

Schmidt number and Soret number causes decrease and increase in concentration

profiles for both slip and no-slip conditions respectively.

Figures (26) and (27) illustrate the effect of Schmidt number (𝑆𝑐) and Soret

number(𝑆𝑟) on the rate of mass transfer. These two figures illustrate that the increase

in Schmidt number and Soret number causes the decrease and increase in 𝜙′(0) for

both slip and no-slip conditions respectively.

Figure (28) explains the effect of skin-friction coefficient 𝑓′′(0) as a function of the

slip parameter for various values of magnetic parameter M. It is clear that skin-

friction decreases rapidly and approaches zero as the slip parameter increases.

Figures (29) and (30) show the rate of heat transfer 𝜃′(0) as a function of the slip

parameters and for various values of magnetic parameter M. We can observe that

increase in slip parameters and causes decrease and increase in the rate of heat

transfer 𝜃′(0) respectively.

Figures (31)-(33) exhibit the rate of mass transfer 𝜙′(0) as a function of the slip

parameters , , and 𝜍 for various values of magnetic parameter M. We can observe

that the rate of mass transfer 𝜙′(0) decreases, as there is increase in slip parameter

and and magnetic parameter M respectively. We can also observe that the rate of

mass transfer 𝜙′(0) increases, with increase in slip parameter 𝜍.


Figure 2. Velocity 𝑓 ′(𝜂) for various

values of k* under the slip and no-slip

boundary conditions.

Figure 3. Temperature q(η) for various



Figure 4. Concentration (η) for various



Figure 5. Velocity 𝑓 ′(𝜂) for various

values of M under the slip and no-slip


Figure 6. Temperatureq(η) for various


boundary conditions

Figure 7. Concentration (η) for various



00.10.20.30.40.50.60.70.80.9

1

0 2 4 6 8 10

f'()

Solid Line = = = 0.0

Broken Line = = =0.1

k*=0.0, 0.2, 0.4, 0.8

S=0.2, M=0.0

00.10.20.30.40.50.60.70.80.9

1

0 2 4 6 8 10

q()



k*=0.0, 0.2, 0.4, 0.8

S=0.2, M=0.0

0

0.5

1

1.5

0 2 4 6 8 10

()



k*=0.0, 0.2, 0.4, 0.8

S=0.2, M=0.0

0

0.5

1

1.5

0 2 4 6

f'()



Pr=7.0;S=0.2;Sc=0.3; Sr=0.3; k=0.2

M=0.0, 1.0, 2.0, 5.0, 10.0

0

0.5

1

1.5

0 5 10

q()

Pr=7.0;S=0.2;Sc=0.3; Sr=0.3; k*=0.2

M=0.0, 1.0, 2.0, 5.0, 10.0

0

0.5

1

1.5

0 5 10

()

Pr=7.0; S=0.2; Sc=0.3;Sr=0.3;k*=0.2

M=0.0, 1.0, 2.0, 5.0, 10.0


Figure 8. Shear stress profiles f ′′(η) for

various values of magnetic field M.

Figure 9. Rate of heat transfer θ′() for

various values of magnetic field M.

Figure 10. Rate of mass transfer ϕ′()

for various values of magnetic field M.

Figure 11. Velocity f ′(η) for various

values of slip parameter

Figure 12. Shear stress f′′(η) for various

values of slip parameter

Figure 13. Temperature θ(η) for various

values of slip parameter .

0

0.5

1

1.5

2

2.5

3

3.5

0 2 4 6

f''()

Pr=7.0;S=0.2;Sc=0.3;Sr=0.3;

k*=0.2; ===0.0

M=0.0, 1.0, 2.0, 5.0, 10.0

-2

-1.5

-1

-0.5

0

0.5

0 2 4 6

q'()

Pr=7.0;S=0.2;Sc=0.3;Sr=0.3;

k*=0.2; ===0.0

M=0.0, 1.0, 2.0, 5.0, 10.0

-0.5

-0.4

-0.3

-0.2

-0.1

0

0 5 10

'()

Pr=7.0;Sc=0.3;Sr=0.3;S=0.2;

k*=0.2; ===0.0

M=0.0, 1.0, 2.0, 5.0, 10.0

0

0.5

1

1.5

0 2 4 6

f'()

Pr=7.0,Sc=0.3,Sr=0.3,S=0.2,

k*=0.2, M=1.0, β=0.2, =0.2

= 0.0, 0.2, 0.5, 1.0, 2.0

0

0.5

1

1.5

0 5 10

f''()

Pr=7.0,Sc=0.3,Sr=0.3,S=0.2,k*=0.2

M=1.0; =0.2; =0.2

= 0.0, 0.2, 0.5, 1.0, 2.0

0

0.2

0.4

0.6

0.8

1

0 2 4

q()

Pr=7.0,Sc=0.3,Sr=0.3,S=0.2,

k*=0.2,M=1.0; =0.2; =0.2

= 0.0, 0.2, 0.5, 1.0, 2.0


Figure 14. Rate of heat transfer θ′(η) for

various values of slip parameter .

Figure 15. Concentration ϕ(η) for


Figure 16. Rate of mass transfer ϕ′(η)

for various values of slip parameter .


values of slip parameter .

Figure 18. Rate of heat transfer θ′(η) for



various values of slip parameter

-1.5

-1

-0.5

0

0.5

0 2 4

q'()

Pr=7.0,Sc=0.3,Sr=0.3,S=0.2,k*=0.2

, M=1.0; =0.2; =0.2

= 0.0, 0.2, 0.5, 1.0, 2.0

0

0.5

1

0 5 10

()

Pr=7.0,Sc=0.3,Sr=0.3,S=0.2,k*=0.2

M=1.0; =0.2; =0.2

= 0.0, 0.2, 0.5, 1.0, 2.0

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0 5 10

'()

Pr=7.0,Sc=0.3,Sr=0.3,S=0.2,k*=0.2

M=1.0;=0.2; =0.2

= 0.0, 0.2, 0.5, 1.0, 2.0

0

0.5

1

1.5

0 5 10

q()

Pr=7.0,Sc=0.3,Sr=0.3,S=0.2,k*=0.2

M=1.0; =0.2; =0.2

= 0.0, 0.2, 0.5, 1.0, 2.0

-2

-1

0

1

0 2 4 6

q'()

Pr=7.0,Sc=0.3,Sr=0.3,S=0.2,k*=0.2M

=1.0; =0.2; =0.2

= 0.0, 0.2, 0.5, 1.0, 2.0

0

0.5

1

1.5

0 5 10

()

= 0.0, 0.2, 0.5, 1.0, 2.0



for various values of slip parameter

Figure 21. Velocity f ′(η) for various

values of suction / blowing parameter S


values of suction/blowing parameter S


various values of suction/blowing

parameter S

Figure 24. Concentration 𝜙(η) for

various values of Sc under the slip and

no-slip boundary conditions

Figure 25. Concentration 𝜙(η) for

various values of Sr under the slip and


-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0 5 10

'()

Pr=7.0,Sc=0.3,Sr=0.3,S=0.2,k*=0.2

M=1.0;=0.2; =0.2

= 0.0, 0.2, 0.5, 1.0, 2.0

0

0.5

1

1.5

0 2 4 6 8

f'()

Pr=7.0,Sc=0.3,Sr=0.3,k*=0.2,

M=1.0, ===0.0

S = -0.4, -0.2, 0.0, 0.2, 0.4

0

0.5

1

1.5

0 2 4 6

q()

Pr=7.0,Sc=0.3,Sr=0.3,k*=0.2,

M=1.0, ===0.0

S = -0.4, -0.2, 0.0, 0.2, 0.4

0

0.5

1

1.5

0 2 4 6 8

()

Pr=7.0,Sc=0.3,Sr=0.3,k*=0.2,

M=1.0, ===0.0

S = -0.4, -0.2, 0.0, 0.2, 0.4

0

0.5

1

1.5

0 5 10

()



Sc = 0.0, 0.1, 0.3, 0.6, 1.0

0

0.5

1

1.5

0 5 10

()



Sr = 0.0, 0.1, 0.3, 0.6, 1.0



for various values of Sc under the slip

and no-slip boundary conditions


for various values of Sr under the slip and


Figure 28. Skin-friction coefficient f ′′(0)

against for various values of M

Figure 29. Rate of heat transfer θ′(0)


Figure 30. Rate of heat transfer

θ′(0) against for various values of M

Figure 31. Rate of mass transfer ϕ′(0)


-1

-0.5

0

0.5

0 5 10

'()



Sc = 0.0, 0.1, 0.3, 0.6, 1.0

-0.5

-0.4

-0.3

-0.2

-0.1

0

0 5 10

'()

Sr = 0.0, 0.1, 0.3, 0.6, 1.0

0

0.5

1

1.5

0 1 2 3

f''(0)

Pr=7.0, Sc=0.3,Sr=0.3,k*=0.2,

S=0.2,=0.2,=0.2

M=0.0, 0.2, 0.5, 1.0

-1.4

-1.2

-1

0 1 2 3

q'(0)

Pr=7.0, Sc=0.3,Sr=0.3,k*=0.2,

S=0.2,=0.2,=0.2

M=0.0, 0.2, 0.5, 1.0

-2

-1.5

-1

-0.5

0

0 1 2 3

q'(0)

Pr=7.0, Sc=0.3,Sr=0.3,k*=0.2,

S=0.2, =0.2,=0.2

M=0.0, 0.2, 0.5, 1.0

-0.5

-0.45

-0.4

-0.35

-0.3

0 1 2 3

'(0)

Pr=7.0, Sc=0.3,Sr=0.3,k*=0.2,

S=0.2, =0.2,=0.2

M=0.0, 0.2, 0.5, 1.0





against for various values of M.

CONCLUSION

The effect of MHD convective boundary layer flow along with heat and mass transfer

over a porous plate embedded in a porous medium has been investigated. The

similarity transformations are used to transform the governing partial differential

equations (PDEs) into a system of nonlinear ordinary differential equations (ODEs).

The resulting system of ODEs is then reduced to a system of first order differential

equations which are solved by shooting procedure using fourth order Runge-Kutta

Method. Effect of various non-dimensional parameters on the fluid flow, heat and

mass transfer characteristics are examined.

The following conclusions are drawn from the present study:

Skin-friction decreases rapidly and approaches zero as the velocity slip

parameter increases.

The rate of heat transfer 𝜃′(0) decreases, with increase in the slip parameter

and magnetic field M. Rate of heat transfer 𝜃′(0) increases, with increase in

the thermal slip parameter .

The rate of mass transfer 𝜙′(0) decreases, as increase in slip parameter ,

and magnetic parameter M and is also observed that the rate of mass transfer

𝜙′(0) increases, with increase in slip parameter 𝜍.

REFERENCES

[1] H.I. Andersson, Slip flow past a stretching surface, Acta Mechanica, 158, 121-125, 2002.

-0.5

-0.45

-0.4

-0.35

-0.3

0 1 2 3

'(0)

Pr=7.0, Sc=0.3,Sr=0.3,k*=0.2,

S=0.2, =0.2,=0.2

M=0.0, 0.2, 0.5, 1.0

-0.45

-0.25

0 1 2 3

'(0)

Pr=7.0, Sc=0.3,Sr=0.3,k*=0.2,

S=0.2, =0.2,=0.2

M=0.0, 0.2, 0.5, 1.0


[2] S.P. Anjalidavi and R.Kandasamy, Effects of chemical reaction, heat and mass

transfer on MHD flow past a semi infinite plate, Z Angew Math Mech, 80, 697–700, 2000.

[3] Asim Aziz, J.I. Siddique, Taha Aziz, Steady Boundary Layer Slip Flow along

with Heat and Mass Transfer over a Flat Porous Plate Embedded in a Porous

Medium, PLoS ONE, 9, e114544, 2014. [4] A. Aziz, A similarity solution for laminar thermal boundary layer over a flat

plate with a convective surface boundary condition, A Commun. Nonlinear Sci. Numer. Simulat, 14, 1064-1068, 2009.

[5] N. Freidoonimehr, M. M. Rashidi and B. Jalilpour, MHD stagnation-point

flow past a stretching/shrinking sheet in the presence of heat

generation/absorption and chemical reaction effects, J Braz. Soc. Mech. Sci. Eng, 38, 1-10, 2015.

[6] K. Gangadhar, N. B. Reddy, and P. K. Kameswaran, Similarity solution of

hydro magnetic heat and mass transfer over a vertical plate with convective

surface boundary condition and chemical reaction, International Journal of Nonlinear Science, 3, 298–307, 2012.

[7] O. D. Makinde, Similarity solution of hydromagnetic heat and mass transfer

aover a vertical plate with a convective surface boundary condition,

International Journal of the Physical Sciences, 5, 700-710, 2010. [8] O.D. Makinde, On MHD heat and mass transfer over a moving vertical plate

with a convective surface boundary condition, Canadian Journal of Chemical Engineering, 88, 983–990, 2010.

[9] Mohammad Mehdi Rashidi, Behnam Rostami, Navid Freidoonimehr and

Saeid Abbasbandy, Free convective heat and mass transfer for MHD fluid

flow over a permeable vertical stretching sheet in the presence of the radiation

and buoyancy effects, Ain Shams Engineering Journal, 5, 901–912, 2014. [10] D. Pal and B. Talukdar, Perturbation Analysis of Unsteady

Magnetohydrodynamic Convective Heat and Mass Transfer in a Boundary

Layer Slip Flow past a Vertical Permeable Plate with Thermal Radiation

Chemical Reaction, Communications in Nonlinear Science and Numerical Simulation, 15,1813–1830, 2010.

[11] Santosh Chaudhary and Pradeep Kumar, MHD Slip Flow past a Shrinking

Sheet, Applied Mathematics, 4, 574-581, 2013. [12] E.M. Sparrow and R.D. Cess, The Effect of Magnetic Field on Free

Convection Heat Transfer,” International Journal of Heat and Mass Transfer,

International Journal of Heat and Mass Transfer, 3, 267-274, 1961.

effect of magnetic field on steady boundary layer slip flow along with heat … · 2017. 3. 9. ·...

Documents